Imbedding classes and 𝑛-minimal complexes

Ummel, Brian R.

doi:10.1090/s0002-9939-1973-0317336-9

Cited by 3 publications

(4 citation statements)

References 7 publications

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“…It is also known that these complexes are minimally nonembeddable, i.e., if we remove from their underlying topological space an arbitrarily small open neighborhood of any point then the resulting space becomes embeddable. However, in higher dimensions, there are not just these two but in fact infinitely many minimally nonembeddable complexes [50,46].…”

Section: Small Nonembeddable Complexesmentioning

confidence: 99%

Minors in random and expanding hypergraphs

Wagner

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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We introduce a new notion of minors for simplicial complexes (hypergraphs), so-called homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes.In this paper, we focus on threshold problems. The basic model for random complexes is the Linial-Meshulam model X k (n, p). By definition, such a complex has n vertices, a complete (k − 1)-dimensional skeleton, and every possible k-dimensional simplex is chosen independently with probability p. We show that for every k, t ≥ 1, there is a constant C = C(k, t) such that for p ≥ C/n, the random complex X k (n, p) asymptotically almost surely contains K k t (the complete k-dimensional complex on t vertices) as a homological minor. As corollary, the threshold for (topological) embeddability of X k (n, p) into R 2k is at p = Θ(1/n).The method can be extended to other models of random complexes (for which the lower skeleta are not necessarily complete) and also to more general Tverberg-type problems, where instead of continuous maps without doubly covered image points (embeddings), we consider maps without qfold covered image points.

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Section: Small Nonembeddable Complexesmentioning

confidence: 99%

Minors in random and expanding hypergraphs

Wagner

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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“…The presentation of the background on the obstruction here is based on the ones in [14], [23] and [19].…”

Section: The Obstruction Over Zmentioning

confidence: 99%

Higher minors and van Kampen's obstruction

Nevo¹

2007

MATH. SCAND.

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We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes H and K and every nonnegative integer m, we prove that if H is a minor of K then the non vanishing of Van Kampen's obstruction in dimension m (a characteristic class indicating non embeddability in the (m − 1)-sphere) for H implies its non vanishing for K. As a corollary, based on results by Van Kampen [20] and Flores [5], if K has the d-skeleton of the (2d+2)-simplex as a minor, then K is not embeddable in the 2d-sphere.We answer affirmatively a problem asked by Dey et. al.[3] concerning topology-preserving edge contractions, and conclude from it the validity of the generalized lower bound inequalities for a special class of triangulated spheres.

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“…It is shown in [1] that X κ is a κ-minimal complex, in the sense that it is not embeddable in R 2κ but each of its proper subcomplexes is embeddable in R 2κ . Cohomological obstructions for embedding in R 2κ are discussed in [5] and [6]. 3 ) is a sphere with 4 handles.…”

Section: Proof Of Lemmamentioning

confidence: 99%

“…This can be shown by computing the Euler characteristic of these spaces and by checking that they are both orientable. I thank the referee for pointing my attention to papers [5] and [6].…”

Section: Proof Of Lemmamentioning

confidence: 99%